Sensitivity Kernels for Local Helioseismology

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Sensitivity Kernels for Local Helioseismology

• Aaron Birch
• NWRA, CoRA Division

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Outline

• Introduction to the forward problem
• Examples of kernels
• Artificial data
• Open questions

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The Forward Problem

• Given a model, e.g. sound-speed, flows want to
  know what to expect for measurements.
• Want linear forward problem necessary for linear
  inversions

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Linear Forward Problems

• Two Steps
• Linearize the dependence of the measurements
  (e.g. travel times) on first order changes in
  wavefield covariance
• Linearize dependence of wavefield covariance on
  changes in the interior of the model (e.g. change
  in sound speed)

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First-Order Change in Cross-Covariance
Use Born approximation to get the sensitivity of
the cross-covariance to perturbations to the
background model

Gizon Birch 2002 ApJ
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Example Kernels
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Line of Sight
J. Jackiewicz and L. Gizon
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Phase-Speed Filtering
Birch, Duvall Kosovichev 2004 ApJ
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Two More Examples
Holography
Time-Distance
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Ring Kernels
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read Woodard 2006 ApJ, coming out soon
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Software

• Currently implemented in matlab
• Now with web interface, very nice !
• This code can do
• Sound-speed kernels for time-distance
• Unperturbed power spectrum and cross-covariance
• Coming Soon flow kernels for holography and
  time-distance
• Requires a background model, source parameters,
  and damping model
• Pretty flexible code (example input file)

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Artificial Data
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Artificial Data

• Use wave propagation simulations to generate
  artificial data (many groups are now working on
  this !)
• Once the machinery is in place will provide a
  very efficient means for solving complicated
  non-linear forward problems
• Explore bias, signal-to-noise ratios, optimize
  measurements, study cross-talk, validation of
  forward and inverse methods

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True velocity
Holography _at_ 6 mHz
Vx
Vy
w/D. Braun. Simulations from Stein Nordlund
also help from Dali Georgobiani.
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Some Open Questions

• For details Gizon Birch 2005 Living Reviews
  Article
• Wave propagation through magnetic regions !
• Source effects ? Parchevsky
• Range of validity of linearizations ?
• Linearization around something other than quiet
  Sun models ? Will this help with Born approx for
  magnetic fields ?
• Currently unknown which kernels are needed only
  for small corrections and which are crucial.
  Simulations will help.
• Line of sight foreshortening. In principle
  need kernels for each position on the disk.
  Likely important for small-scale flows and
  sound-speed variations.